174 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



This equation leads to 



/ ^ J . 2fltt l 

 ( " + 1 > + TJ 



2ws' 



There exists then a certain law of depth, depending on the period, for which the 

 tide will always be similar in type to the disturbing potential which produces it. 

 This law of depth is expressed by the formula 



If we supposes that X = 2w rigorously, it reduces to 



so that the depth will be a maximum at the equator, and will gradually decrease on 

 passing away from the equator to zero at the poles.* 



For other values of X the formula for h will make the depth negative at some parts 

 of the surface unless / is positive and X > 2&>. The latter condition does not occur 

 with any of the leading tidal constituents^ but it would hold good in the case of the 

 semi-diurnal tides due to a satellite whose motion in its orbit was retrograde. If 

 however we neglect the mutual attraction of the waters, the theorem under dis- 

 cussion may be supposed to apply to an ocean covering that part of the surface over 

 which h is positive, the remaining parts of the surface being supposed to consist of 

 continents. When / is positive, these continents must, for the lunar semi-diurnal 

 tides, reduce to small circurnpolar islands, while for the same tides when I is nega- 

 tive they will cover the whole globe with the exception of two small seas surrounding 

 the poles. 



For the diurnal-tides, the shores must coincide with parallels of latitude approxi- 

 mately 30 north and south of the equator, while for the tides of long period the 

 appropriate forms of sea will be bounded by two parallels nearly coincident with the 

 equator. A change in the sign of I in all cases involves an interchange between the 

 seas and the land. 



It should be noticed that the formulae (12) make U, V infinite at the points where 



p- rfc -, that is, at the shores. This indicates that the neglect of the squares of 



the velocities is not allowable in the neighbourhood of the coasts no matter how small 

 the amplitude of vibration may be, and seems to point to the existence of "breakers" 

 as an essential accompaniment of the tides. 



* Gf. LIMB, ' Hydrodynamics,' 213. 



t There will be small tidal constituents depending on higher powers of the moon's parallax for 

 which X exceeds 2w. Gf. DARWIN, " Harmonic Analysis of Tidal Observations'." ' Brit. Assoc. Report,' 

 1883 (Sonthport), 3. 



